Calculus Examples

$lim_{x \to \infty} \frac{x^{2} - 9}{x (x^{2} + 1)}$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$lim_{x \to \infty} \frac{x^{2} - 9}{x \cdot x^{2} + x \cdot 1}$

Step 1.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.2.1

Multiply $x$ by $x^{2}$ .

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Step 1.2.1.1

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{x^{2} - 9}{x^{1} x^{2} + x \cdot 1}$

Step 1.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to \infty} \frac{x^{2} - 9}{x^{1 + 2} + x \cdot 1}$

Step 1.2.2

Add $1$ and $2$ .

$lim_{x \to \infty} \frac{x^{2} - 9}{x^{3} + x \cdot 1}$

Step 1.3

Multiply $x$ by $1$ .

$lim_{x \to \infty} \frac{x^{2} - 9}{x^{3} + x}$

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{3}$ .

$lim_{x \to \infty} \frac{\frac{x^{2}}{x^{3}} + \frac{- 9}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 3.1.1.1

Multiply by $1$ .

$lim_{x \to \infty} \frac{\frac{x^{2} \cdot 1}{x^{3}} + \frac{- 9}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}}}$

Step 3.1.1.2

Cancel the common factors.

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Step 3.1.1.2.1

Factor $x^{2}$ out of $x^{3}$ .

$lim_{x \to \infty} \frac{\frac{x^{2} \cdot 1}{x^{2} x} + \frac{- 9}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}}}$

Step 3.1.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x^{2} \cdot 1}{x^{2} x} + \frac{- 9}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}}}$

Step 3.1.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{1}{x} + \frac{- 9}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}}}$

Step 3.1.2

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{\frac{1}{x} - \frac{9}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}}}$

Step 3.2

Simplify each term.

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Step 3.2.1

Cancel the common factor of $x^{3}$ .

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Step 3.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{1}{x} - \frac{9}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}}}$

Step 3.2.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{1}{x} - \frac{9}{x^{3}}}{1 + \frac{x}{x^{3}}}$

Step 3.2.2

Cancel the common factor of $x$ and $x^{3}$ .

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Step 3.2.2.1

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{\frac{1}{x} - \frac{9}{x^{3}}}{1 + \frac{x^{1}}{x^{3}}}$

Step 3.2.2.2

Factor $x$ out of $x^{1}$ .

$lim_{x \to \infty} \frac{\frac{1}{x} - \frac{9}{x^{3}}}{1 + \frac{x \cdot 1}{x^{3}}}$

Step 3.2.2.3

Cancel the common factors.

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Step 3.2.2.3.1

Factor $x$ out of $x^{3}$ .

$lim_{x \to \infty} \frac{\frac{1}{x} - \frac{9}{x^{3}}}{1 + \frac{x \cdot 1}{x \cdot x^{2}}}$

Step 3.2.2.3.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{1}{x} - \frac{9}{x^{3}}}{1 + \frac{x \cdot 1}{x \cdot x^{2}}}$

Step 3.2.2.3.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{1}{x} - \frac{9}{x^{3}}}{1 + \frac{1}{x^{2}}}$

Step 3.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} \frac{1}{x} - \frac{9}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}}}$

Step 3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{9}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}}}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{0 - lim_{x \to \infty} \frac{9}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}}}$

Step 5

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{0 - 9 lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}}}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$\frac{0 - 9 \cdot 0}{lim_{x \to \infty} 1 + \frac{1}{x^{2}}}$

Step 7

Evaluate the limit.

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Step 7.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{0 - 9 \cdot 0}{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 7.2

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

$\frac{0 - 9 \cdot 0}{1 + lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{0 - 9 \cdot 0}{1 + 0}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $- 9$ by $0$ .

$\frac{0 + 0}{1 + 0}$

Step 9.1.2

Add $0$ and $0$ .

$\frac{0}{1 + 0}$

Step 9.2

Add $1$ and $0$ .

$\frac{0}{1}$

Step 9.3

Divide $0$ by $1$ .

$0$